The Keller-Segel system with logistic growth and signal-dependent motility

The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, 0=\Delta v+ u-v,& x\in \Omega, ~~t>0,\\ u(x,0)=u_0(x), & x\in \Omega, \end{cases} \end{equation} with homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \R^2$ with smooth boundary, where the motility functions $\gamma(v)$ and $\chi(v)$ satisfy the following conditions \begin{itemize} \item {\color{black}$(\gamma,\chi)\in [C^2[0,\infty)]^2$} with $\gamma(v)>0$ and {\color{black} $\frac{|\chi(v)|^2}{\gamma(v)}$ is bounded for all $v\geq 0$.} %for all $v\geq 0$ and $\lim\limits_{v\to\infty}\frac{|\chi(v)|^2}{\gamma(v)}$ exists. \end{itemize} By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref{0-1} with $\mu>0$ for any $u_0 \in W^{1, \infty}(\Omega)$. Then based on a Lyapunov function, we show that all solutions $(u,v)$ of \eqref{0-1} will exponentially converge to the unique constant steady state $(1,1)$ provided $\mu>\frac{K_0}{16}$ with $K_0=\max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)}$.